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Infinite Powers: How Calculus Reveals the Secrets of the Universe

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Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not abo Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS. As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.


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Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not abo Without calculus, we wouldn’t have cell phones, TV, GPS, or ultrasound. We wouldn’t have unraveled DNA or discovered Neptune or figured out how to put 5,000 songs in your pocket. Though many of us were scared away from this essential, engrossing subject in high school and college, Steven Strogatz’s brilliantly creative, down‑to‑earth history shows that calculus is not about complexity; it’s about simplicity. It harnesses an unreal number—infinity—to tackle real‑world problems, breaking them down into easier ones and then reassembling the answers into solutions that feel miraculous. Infinite Powers recounts how calculus tantalized and thrilled its inventors, starting with its first glimmers in ancient Greec e and bringing us right up to the discovery of gravitational waves (a phenomenon predicted by calculus). Strogatz reveals how this form of math rose to the challenges of each age: how to determine the area of a circle with only sand and a stick; how to explain why Mars goes “backwards” sometimes; how to make electricity with magnets; how to ensure your rocket doesn’t miss the moon; how to turn the tide in the fight against AIDS. As Strogatz proves, calculus is truly the language of the universe. By unveiling the principles of that language, Infinite Powers makes us marvel at the world anew.

30 review for Infinite Powers: How Calculus Reveals the Secrets of the Universe

  1. 5 out of 5

    Peter Mcloughlin

    I feel bad for kids who do ordinary arithmetic in grade school. For me, the math doesn't get interesting until you get above the Calculus line. Calculus with its dealings with the continuum is the first real taste of the infinite. If you stick around for Calculus I bet you would want more. Strogatz in this book shares some of the excitement we have once the math gets weird. In some places, math can be hallucinogenic. I encourage people to look beyond grade school stuff which is way more interest I feel bad for kids who do ordinary arithmetic in grade school. For me, the math doesn't get interesting until you get above the Calculus line. Calculus with its dealings with the continuum is the first real taste of the infinite. If you stick around for Calculus I bet you would want more. Strogatz in this book shares some of the excitement we have once the math gets weird. In some places, math can be hallucinogenic. I encourage people to look beyond grade school stuff which is way more interesting especially if you have been disappointed by math. and totally unrelated a Pink Floyd mandelbrot zoom for Hippies of a Mathematical platonist bent. https://www.youtube.com/watch?v=b7JbJ... oops accidentally inserted the wrong video on AI which is also interesting but not intentional. https://www.youtube.com/watch?v=1wAgB...

  2. 4 out of 5

    Ryan Boissonneault

    Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a sub Calculus is one of those subjects that is so complicated that most people not only don’t understand it, they don’t even know what it is that they don’t understand. But that’s unfortunate, because calculus is one of humanity’s most impressive achievements, an accomplishment that unlocks the secrets of the universe and delivers our most profound and useful technology, from radio and television to GPS navigation and MRI imaging. Calculus is the main protagonist in the story of science, and is a subject every educated person should understand at least conceptually. Fortunately, you don’t have to trudge through a thousand-page textbook to appreciate the story and power of calculus. Steven Strogatz, in his latest book Infinite Powers, has provided a clear, concise, and fascinating tour of the subject. In fact, if you don’t understand what calculus is all about after reading this book, then the prospects are not great that you ever will. There is simply no better, clearer presentation of the ideas available. Strogatz uses metaphors, illustrations, stories, and examples to guide the reader through the most difficult concepts. While this is not an easy read, it is as reader-friendly as possible; remember, you’re tackling the most sophisticated branch of mathematics, the underlying logic of all science, and a subject that the sharpest mathematical minds in history had to grapple with for thousands of years. As Strogatz explains, calculus is difficult because it’s tackling the most difficult problems humans encounter, problems that necessitate complex equations, notation, and mathematical manipulation. But behind this computational complexity lies an obsession with simplicity, with breaking down hard problems into easier parts. The special innovation of calculus, as Strogatz explains, is that problems are broken down into infinitely small and manageable parts and then recombined back into the whole. So what is calculus, exactly? It’s easier to describe calculus by the types of problems it solves than by standard mathematical definitions. When most people hear terms like “infinite series,” “limits,” “derivatives,” and “integrals,” they lose sight of the bigger picture of what calculus is trying to accomplish. One type of problem calculus can solve is the area under a curved surface. Area is typically quite easy to solve for shapes with straight lines. For rectangles, for example, the area is simply length times width. But what about for shapes with curves where the slope is constantly changing? There is no simple formula to calculate the area in this situation. You could approximate the area by overlaying rectangular objects over the curved shape (as shown below), but this would only be an approximation as the rectangles would not fit exactly in the curved shape. However, as you made the rectangles smaller (and increased their number) the fit inside the curved shape would keep improving and the approximation would keep getting more accurate. Since you can always keep dividing a number in half (you can always make a number larger or smaller), you can add an infinite number of smaller rectangles into the curved shape. You can never complete this process (which is why the concept of “completed infinity” is logically incoherent), but you could potentially keep adding rectangles forever, which is logically coherent and shows the difference between “completed infinity” and “potential infinity.” As you increase the number of rectangles, you get closer and closer to the area, which is the limit of the infinite series. The area of the curved shape becomes the sum of the infinite series of rectangles. Calculus is the set of equations and procedures to carry out this calculation precisely. Calculus can also solve problems of motion. Straight-line motion at constant velocity is easy. If you know the speed of an object, then the distance traveled is simply speed times time. But how can you calculate the trajectory of, say, a planet, that not only continuously changes direction in orbit around the sun but that also speeds up or slows down depending on its distance from the sun? This is not so easy, but is solved in a similar way by breaking down the trajectory into infinitely smaller parts and then summing the series. Calculus provides the procedures and notation to carry this out in the most efficient way. You’ll notice that both examples above solve for problems where some quantity is continuously changing. That means that calculus can solve any problem that involves a quantity that is continuously changing, like the spread of a virus, population growth, or continuously compounding interest in finance. Even without understanding the specific calculations, it’s amazing to contemplate the fact that we can harness the power of infinity to calculate with precision the area under any curved surface, the dynamics of any continuously changing variable, and the trajectory of any object anywhere in the universe! Of course, this brief sketch is only a description of the subject in its simplest terms; there is much more to the subject and the mechanics of the calculations gets incredibly complex. If you’re interested in diving deeper into the subject, with examples and proofs, Strogatz delivers a nice mixture of pure mathematics, practical examples, and a history of the personalities behind the development of calculus. Of particular interest for me was Strogatz’s solution of Zeno’s Achilles and the Tortoise paradox, a solution that finally made sense to me (in brief, the solution is that an infinite amount of steps can be completed in a finite amount of time). The Power of Human Cooperation If you find calculus near impossible to learn, you won’t be happy to know that Isaac Newton invented the subject before he turned 25. But you might find some solace in the fact that Newton did little else; he had few friendships and no romantic relationships, so he had all the time in the world to devote to numbers and experiments. Newton also couldn’t have done it alone. He was exactly right when he said that he was able to see further by “standing on the shoulders of giants.” As Strogatz explained: “But again, he [Newton] couldn’t have done any of this without standing on the shoulders of giants. He unified, synthesized, and generalized ideas from his great predecessors: He inherited the Infinity Principle from Archimedes. He learned his tangent lines from Fermat. His decimals came from India. His variables came from Arabic algebra. His representation of curves as equations in the xy plane came from his reading of Descartes. His freewheeling shenanigans with infinity, his spirit of experimentation, and his openness to guesswork and induction came from Wallis. He mashed all of this together to create something fresh, something we’re still using today to solve calculus problems: the versatile method of power series.” There are at least two lessons here; first, knowledge grows exponentially, not linearly, and there is no limit to what can be discovered. By standing on the shoulders of giants, each generation can build on the developments of the past, as Einstein was able to do by rejecting Newton’s ideas of space and time as absolute. Holding a person, idea, or generation in complete reverence inhibits progress, as when we followed Aristotle for 1,500+ years and maintained the belief that the earth was stationary. The best book I’ve read that elaborates on this point is The Beginning of Infinity: Explanations That Transform the World by the physicist David Deutsch. Second, calculus demonstrates the power of human cooperation. No single mind could have developed calculus from scratch. People of diverse origin and circumstance collaborated to find solutions to common, tangible problems, because they didn’t waste their time thinking about arbitrary human divisions and other products of pure imagination, like religion. Newton borrowed from ancient Greek geometry, French analytic geometry, the Indian decimal system, and Arabic algebra. As a result, he discovered the mathematical logic and underlying laws of nature that applied equally to objects anywhere in the universe, thus uniting the entire cosmos. This universality, as Strogatz recognized, sparked the beginning of the Enlightenment. A final point: in the concluding chapter, Strogatz describes Richard Feynman’s quantum electrodynamics (QED) theory, which, by using calculus, describes the quantum interaction of light and matter. Physicists use the theory to make predictions about the properties of electrons and other particles. As Strogatz said, “by comparing those predictions to extremely precise experimental measurements, they’ve shown that the theory agrees with reality to eight decimal places, better than one part in a hundred million.” This means that QED is the most accurate theory anyone has ever devised about anything. A prediction with an accuracy of 8 decimal places is like, using Strogatz’s example, planning to snap your fingers exactly 3.17 years from now down to the second, without the help of a clock or alarm. As Strogatz further explains: “I think this is worth mentioning because it puts the lie to the line you sometimes hear, that science is like faith and other belief systems, that is has no special claim on truth. Come on. Any theory that agrees to one part in a hundred million is not just a matter of faith or somebody’s opinion. It didn’t have to match to eight decimal places.” You will also often hear that science can’t determine right and wrong actions, which in some sense is correct, but misses the point. The moral element of science does not lie in any particular factual claim; it lies in the orientation to forming beliefs. The scientific mindset is not about clinging on to and forming your identity around a set of unalterable beliefs. The scientific mindset is about curiosity, orientation to discovering truth, intellectual integrity, and revising beliefs in the face of new evidence. It’s also, as I believe calculus shows, about the recognition of the power of human cooperation and the pursuit of knowledge as a collective human endeavor.

  3. 5 out of 5

    Eric

    TL;DR In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended. Review cross-posted at Primmlife.com Review When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation moving I agree, and while I do lik/>/>Review TL;DR In Infinite Powers,Dr. Steven H. Strogatz teaches us how to use our microwaves to calculate the speed of light. I’m not kidding. That’s all the recommendation this book needs. Highly Recommended. Review cross-posted at Primmlife.com Review When I tell people that I’m an engineer, my wife likes to follow up that comment with, “He does math all day long.” A common response is, “Oh, you must really like math. I didn’t enjoy it in [insert level of schooling here].” To keep the conversation moving I agree, and while I do like math, I didn’t always. Until I started studying calculus, math bored me. Algebra existed as a set of rules; geometry, though my introduction to proofs, seemed too abstract. But when I first solved a derivative, my indifference turned to frustration and intrigue. My plan to take only enough math to get an engineering degree changed to a serious contemplation of switching career paths to applied mathematics (with an eye towards physics grad degrees). Ultimately, I stuck with the engineering curriculum but ended up studying higher level mathematics, and to this day, I still read about and love math. Part of my studies now involves going back and filling in what I missed during previous years. One of the voices to which I turn is Dr. Steven Strogatz, and his latest book, Infinite Powers, fills in details about calculus that I lacked. His explanations don’t rely on the familiar equations but, instead, root themselves in history, in logic, and in excellent prose. Infinite Powers transforms calculus from equations into meaning. The Story In Infinite Powers, Dr. Strogatz starts with Archimedes from ancient Greece and carries on to some of today’s most unique challenges. It is the story of calculus told as a continuum of human learning. Often, the public thinks that scientific breakthroughs happen when lone geniuses discover something new, but in reality discoveries occur when people improve upon the work of others. In Infinite Powers Dr. Strogatz traces the methods Archimedes used to Newton and Leibniz, who are the inventors or discoverers of calculus. Along the way, we learn about contributions from Fermat, Galileo, Descartes, Arabic, and Chinese mathematicians. But we don’t stop at the discovery era. Infinite Powers continues on to Fourier and Sophie Germain. We even get to see how calculus is being used today to treat HIV patients, to create microwaves, and, near to my own heart and pocketbook, help the 787 fly. The Writing Math possesses a strong language of its own rooted in symbols and logic. While I view this as a strength, I also know others view the equations, Greek letters, and symbols to be inscrutable. Others have said that math texts tend to be dry reads. For anyone who thinks this, Infinite Powers is the book for you. While equations do exists, they are few. Dr. Strogatz takes the time to explain, in detail, what each of the symbols means. But the majority of the book reads more like a history text than a mathematical treatise. While it doesn’t spoon feed the reader, it doesn’t bog down in jargon. Clarity and simplicity are the descriptors I have already used talking about this book with friends. Dr. Strogatz does an excellent job describing what the math is actually doing. The reader will NOT be able to do any calculus after reading it, but he/she will understand how powerful a tool it is. There are graphs and pictures throughout the book. In my advanced reader’s copy (ARC), the graphs didn’t show up. So, I cannot speak to their quality; however, with my background and the detail of Dr. Strogatz’s descriptions, I could picture what his intent was with the graphs. That should be an indicator of success for the prose of this book. Ugh, Math, Really? Bear with me here as I get on my soapbox for a minute. One of the other responses that I get when I’m introduced as an engineer is, “You must be really good at math.” And compared to most people, yes, I am good at math. But I’m good at math for one reason only, I’ve been practicing it in one form or another for the last 23 years. In the martial arts, there’s a saying that a black belt is simply a white belt who didn’t quit. To me, that’s all that math is. I’m good at math and calculus because I didn’t quit doing math. The general public often thinks that math requires a certain mindset or, even, a certain person. No, it requires practices and tenacity. The reason that I stuck with math is because of teachers in high school that showed the same enthusiasm that Dr. Strogatz shows in this book. Teachers and professors who care that students understand a subject make this world a better place. After reading Infinite Powers, I have no doubt Dr. Strogatz is a teacher than inspires students. I can’t help but wonder what could happen if a book like this gets into the hands of someone who thinks they have to be good at math to understand it. Math as Art Though we use math in the sciences, I’ve come to view it more as an art. The mathematician, engineer, chemist, or whoever must know and understand the tools math gives us in order to solve problems, and like a painter picks and chooses the right brush to add to the painting, the problem solver picks and chooses the correct mathematical tool. It’s a creative process that, instead of being hung in a gallery or museum, zips down the road, flows through our veins, or launches a satellite into space. Dr. Strogatz demonstrates the versatility and creativity that we are capable of when using calculus. Whether putting satellites in space or determining how viruses spread, calculus is a tool for delving into nature’s mysteries. Infinite Powers stirred that creative sense, that feeling of awe at being able to see into the universes internal mechanisms. At the same time, it reminded me of the ingenuity of the human animal to seek out and explore the world around us. Dr. Strogatz conveys the beauty that one can find in math, and I felt that thrill of discovery again as I read this book. Infinity Originally, I requested this book because I thought it was about infinity. That mathematical concept that looks like an 8 fell asleep, ∞. Instead, it was about calculus; so, I went into the first few chapters with the wrong expectations. Dr. Strogatz discusses infinity but not enough to satisfy me. And throughout the book, he does reference back to the topic of infinity, but it feels more like a forced attempt to tie the later chapters to the theme. I’m still hoping that Dr. Strogatz gives us a book about infinity in the same detail and manner that he gave us a book about calculus. Conclusion Dr. Steven Strogatz’s Infinite Powers details the history and development of calculus. Dr. Strogatz’s ability to relate complex mathematical concepts in clear and precise language is at peak form in this book. For anyone curious about calculus, this book provides answers in delightful, easy to understand prose that will awaken your curiousity.

  4. 4 out of 5

    Tam

    I certainly wish I had read this book while in high school or college. We grilled all the basic technical parts of calculus and yet unsure what was the point. Well certainly you don't need to know it if you do not work in science and research. Life can go on just as well. But being able to appreciate the beauty of it is an added bonus. And then who knows, seeing that beauty could change the path you take in life. Strogatz takes the same approach with his earlier pop science book, The I certainly wish I had read this book while in high school or college. We grilled all the basic technical parts of calculus and yet unsure what was the point. Well certainly you don't need to know it if you do not work in science and research. Life can go on just as well. But being able to appreciate the beauty of it is an added bonus. And then who knows, seeing that beauty could change the path you take in life. Strogatz takes the same approach with his earlier pop science book, The Joy of x, i.e. trying his best to explain calculus via intuition, allegories, and graphs. It works up to some extent. Some concepts require a lot more, but as the author tries to avoid discussing the technicals, the discussion becomes a bit shallow. That is still fine, this is not a textbook. What I'm a bit uneasy about is Strogatz's reverence for calculus to the point of being religious. Maybe I have taken it wrong, taken his comparison/allegory too much at face value when he cites Feymann over and over again that it is the language of God. Just as the title says, Strogatz argues that calculus is so special, nature acts according to the rules of calculus, everything can be described using these types of equations. Perhaps I've taken calculus for granted. Of course it is behind everything. Modern science is built upon math and ultimately one of the most powerful tools ever invented. Yes, calculus is a tool, a language. It is a lens of looking at the world and of course whatever picture one sees would be described in that medium. Is it so surprising? Could there be another language that provides a different lens to see the world? I'm an illiterate here, I have no idea. My take is that calculus has been so successful to deal with so many problems of interest, and there are so many things to extend and develop, that one isn't incentivized to go through all the pain to develop another language to compete with it. But I would prefer to think that there are possible options out there, so that calculus is not necessary the one (and only) language of "God." Strogatz also emphasizes that calculus is behind everything, behind all sorts of important equations and inventions in human history, driving progress and success. It is true that calculus is used, but I feel that the writing is tilted towards overstressing the importance of calculus and downplaying all other types of ingenious ideas and inventions that work together. Perhaps one needs to exaggerate to attract attention. It is the way of writing not the idea. And of course any language has its limitation. Strogatz gives a lot of praises and does not so much tell readers what are the stuffs that calculus cannot deal with. Sure, towards the end he does say there is a limit to what mathematicians can solve, such as nonlinearity. He hints at the need for another type of method to approach such problems, such as that from Poincare. The discussion is not as comprehensive as I would like. Don't get me wrong, I love calculus. I'm more of an abstract thinker and do not need much to be convinced that calculus a wonderful and elegant tool. Yet I have problems with the writing. I prefer writings with more nuance. I ended up looking forward to the bits talking about mathematicians and their private lives. But Professor Strogatz is not a historian, so I was naturally also not satisfied either.

  5. 4 out of 5

    Al Bità

    For an ordinary layperson, this is perhaps the most accessible history of the development of Calculus one could hope for. In easily readable language Strogatz has provided a fascinating narrative covering the ideas behind Calculus, its history from the earliest Greek mathematicians, its “dismissal” from the formal geometric/mathematical canon for some two thousand years, until its resurgence in the 17th-c with the work of Newton and Leibnitz, and on to its amazingly extensive application to just For an ordinary layperson, this is perhaps the most accessible history of the development of Calculus one could hope for. In easily readable language Strogatz has provided a fascinating narrative covering the ideas behind Calculus, its history from the earliest Greek mathematicians, its “dismissal” from the formal geometric/mathematical canon for some two thousand years, until its resurgence in the 17th-c with the work of Newton and Leibnitz, and on to its amazingly extensive application to just about every sphere of activity in modern civilisation. Some knowledge of basic mathematics is required, but not much more than that: Strogatz is more concerned with explaining what Calculus is all about, and pointing out that certain precise questions about specific physical problems can only be best answered by its application. Calculus is the most accurate mathematical tool ever developed for answering such questions. Some things I particularly liked about this lie in the conceptual (philosophical?) matters that are implied or suggested: the idea, for example, that the modern understanding of the Space-Time continuum is maintained (for me, the “problem” between the continuum idea and the idea of infinitesimals remains, and probably should remain so); and the fact that, no matter how “infinitesimal” the divisions of differential calculus are made, there will always remain “a little bit” extra that is unaccounted for. The latter is difficult enough when dealing with the interrelationship of two “entities” or “bodies”, let alone the far more complicated problem relating to three (or more?) such entities, so it remains a potentially intriguing matter. While the cynics among us might consider this enough to disparage its absolute applicability, Calculus remains our most successful, accurate and practical measurement tool so far. I like the idea that there is still the possibility for new approaches to be made: but that being said, the current “solution” to such errors will still most probably be found through the re-application and refinements of its processes. An ironic example of this relates to Strogatz’s use of the Boeing 787 Dreamliner aeroplane as a proud example of the achievements made under the auspices of this form of mathematics. This may very well be the case, but at the same time, some bad reports coming in 2019 about Boeing’s 737 MAX line of planes suggested that, among other matters, computer design flaws were partly responsible for two of these planes crashing with a total loss of 364 lives: a sober reminder that the more intricate the maths, the more care needs to be taken in its applications and consequences in the physical world. Even so, it is Calculus which is at the heart of most of the technological developments we already enjoy, and it will undoubtedly have much more to contribute, particularly in the areas of medicine, society and politics in the years to come. This book will at least provide a basic understanding of what it is that has become indispensable in our societies, whether we like it or not.

  6. 4 out of 5

    Athan Tolis

    I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here. So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the I need to psyche myself up to do some math for work. And I have a math sherpa and I arranged to meet him so he can take me through the paper I must tackle. But I’m old and only really remember my high school math well, so there is a genuine task at hand here. So I duck and dive between the paper and my notes from my MSc thesis from at least fifteen years ago and I work out the answer to lesser problems and I write out my questions for my sherpa and I also need to be thinking math the whole time; I need to be in a mood, basically. That’s the task. So I did the sensible thing and went on a bit of a binge and bought a whole bunch of popular math books in one go to read in the tube. “Infinite Powers” I read first, because it looked like it would not challenge me at all and it gets good writeups. It’s bloody awesome! It’s more than an anthology of results and it’s more than a series of mini-portraits of mathematicians, it’s almost got a plot. Surprisingly often, even the obligatory corny applications of the math are (somewhat) related to what the author’s talking about. Huge caveat: I knew both the math and even many of the stories upfront, so perhaps it’s not very well explained. I have no way of knowing. But I bet you it is. Perhaps not well enough that you could hope to learn calculus from here, Jordan Ellenberg’s praise on the back cover notwithstanding. (For that I can refer you to “Quick Calculus” by twin gods Kleppner and Ramsey.) But probably well enough to be a companion to anybody taking calculus for the first time. Steven Strogatz had me from “hello,” of course, because he starts with the Greeks, on whom he lavishes immense praise. He could have left it there and I’d still be basking in the warm glow of my ancestors’ work. Needless to say, it does not stop there, he takes you from them to Fermat and Descartes, before introducing you to Newton and Leibniz, a couple words on Fourier and from him straight to Einstein, taking special care to erase all traces of evil men like the unspeakable inventor of delta-epsilon proofs. You won’t find the C-word here. So there’s a massive hole in the nineteenth century, somewhere, but I’m sure you can buy another book to find out about that. Here you’ll discover a decent definition of e, an intuitive explanation of general relativity, the common cause of death of Leibniz and Newton, a fun game to play with your microwave oven, the first and second derivative of the sine wave, the dimension of the three-body problem, a strong defense of infinitesimals, WHAT’S NOT TO LIKE? Enough from me, I’ll now go buy some extra copies for a few boys and girls I know. If one of them likes it, my job is done. Oh, sorry, one more thing. About the plot: it’s a history of how mathematicians throughout time have sliced hard problems into infinite infinitely-thin slices where the problem has a clear answer and then dealt with infinity to sum up the solutions to the easy problem in order to come up with an answer to the hard problem. Whenever you do that, you’re doing calculus, you’re putting together the answer granule by granule.

  7. 4 out of 5

    Rama

    This book does not make calculus interesting Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is This book does not make calculus interesting Calculus is widely perceived as important part of science in understanding basic laws of physics. But it also has important applications in advanced physics; relativity and quantum mechanics, cosmology, astronomy, biology, chemistry, medicine, geology, ecology and in everyday life. In this book, the author discusses calculus as catch-as-catch-can story in an historical context without giving some ideas of how calculus helped physics to evolve. This is not a recipe book and at the same time it is not overwhelming. But in the absence of clear mathematical methods or its applications, this is a slapdash story that does not make calculus interesting.

  8. 5 out of 5

    Martynas Petkevičius

    Infinite Powers won't teach you calculus but it'll gently familiarise you with the subject without oversimplifying it. You'll finish the book wanting to dive into the actual maths.

  9. 5 out of 5

    Roberto Rigolin F Lopes

    A few centuries ago some clever people noticed that nature is in an ever-changing state, notably Galileo (1564-1642) studying objects in free fall and Kepler (1571-1630) studying the motion of planets around our sun. Then Newton (1643-1727) and Leibniz (1646-1716) invented a mathematical tool to get closer and closer to the changing system at hand. Steven did a great job explaining how Calculus uses divide-and-conquer to the extreme taming infinity to describe the universe. It changed civilizati A few centuries ago some clever people noticed that nature is in an ever-changing state, notably Galileo (1564-1642) studying objects in free fall and Kepler (1571-1630) studying the motion of planets around our sun. Then Newton (1643-1727) and Leibniz (1646-1716) invented a mathematical tool to get closer and closer to the changing system at hand. Steven did a great job explaining how Calculus uses divide-and-conquer to the extreme taming infinity to describe the universe. It changed civilization; this book travels from Archimedes (-212) computing pi to today’s design of airplanes. And Calculus is still evolving like a living organism after an explosion of diversity to explain CHANGE everywhere. For example, Einstein (1879-1955) used Calculus to play with space (say x, y, z) and time, at least four things changing at the same time.

  10. 4 out of 5

    Bookjazzer2010

    4++ Wow! I wish I had had this book back in college 50+ years ago. Who knew calculus could be this interesting to read about. I only endured my class with a boring professor. No joy then, but I thoroughly enjoyed this book. All the applications of calculus through the years were fascinating. Amazing stories about the scientists and mathematicians. Very easy to read and understand. I only needed to back up to reread a few paragraphs.

  11. 5 out of 5

    Lemar

    "Dividing by zero summons infinity in the same way that a Ouia board supposedly summons spirits from another realm. It's risky. Don't go there." This sentence gives a good idea of the fun and rigor that Steven Strogatz brings to this book that explains what the big deal is to people who, let's face it, are unlikely to learn calculus. "The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five hundred year story of calculus." By "Dividing by zero summons infinity in the same way that a Ouia board supposedly summons spirits from another realm. It's risky. Don't go there." This sentence gives a good idea of the fun and rigor that Steven Strogatz brings to this book that explains what the big deal is to people who, let's face it, are unlikely to learn calculus. "The desire to harness infinity and exploit its power is a narrative thread that runs through the whole twenty-five hundred year story of calculus." By weaving examples of what's so damn useful about calculus with stories of great minds and the problems they overcame personally and mathematically, Strogatz wrote a readable, yes, fun book about math. His is not a big ego, he gives pioneers their due and busts some myths along the way, "the Pythagorean theorem did not originate with Pythagoras, it was known to the Babylonians for at least a thousand years before him." We learn about the Chinese genius Liu Hui who improved on Archimedes's method for calculating pi as well as Zu Chongzhi who pushed the study of polygons further than anyone before him. Strogatz has a special affinity for the Sicilian Greek Archimedes. The Hindu contribution is enormous and seems worth an entire book on its own. The Arabic scholar Al-Hasan Ibn al-Haytham gets recognized as one of the many giants upon whose shoulders Newton and Leibniz stood. We don't stop there but in several great chapters we learn about the absolute latest breakthroughs and applications of calculus, to Quantum Electrodynamics and Chaos theory, medicine and many other non-linear problems. Stretching my inelastic brain to the the snapping point, he discusses Eintstein's partial derivative theories. "matter tells space-time how to curve, while curvature tells matter how to move. The dance between them makes the theory nonlinear." I love that sentence! Strogatz is a true writer, an artist with words who, happily for me, applies that skill to explaining his profession, and love, calculus.

  12. 5 out of 5

    Yvo Hunink

    If I had only read this book before I got calculus in university. I would have maybe actually understood what I was doing. I advise anyone that is doing, or will do, calculus, to use this book as a great additional reading that will probably save you some hours in the end. I got this lovely book from my dear friend Kristina, thanks! Calculus is just not a set of mathematical tricks we use to calculate stuff. The unfolding of proofs, the building on concepts and the progression of the If I had only read this book before I got calculus in university. I would have maybe actually understood what I was doing. I advise anyone that is doing, or will do, calculus, to use this book as a great additional reading that will probably save you some hours in the end. I got this lovely book from my dear friend Kristina, thanks! Calculus is just not a set of mathematical tricks we use to calculate stuff. The unfolding of proofs, the building on concepts and the progression of the logics of change. It is the core of our collective understanding, to which we owe much of our well-being. From Archimedes, to Galileo, to Newton and now... to Poincaré. I was positively surprised that towards the end of the book, one of the founding fathers of the non-linear, chaotic and complex realm was mentioned, for his expansion of the pendulum dynamics away from the ordered domain. While I was not looking for anything on complex systems in this book, apparently it is hunting me ;). Very good to hear that Strogatz views this field as an interesting topic for calculus in the 21st century. However, that's also where Strogatz lost the 5 stars. He missed the opportunity to explicitely define fractals as the new geometric shape to be studied and fractal dimensions as the go-to derivatives, so that we can maybe find the mysteries of emergence. The last chapter, for me, could have been expanded into 3 more chapters. Especially more about AI and how we use calculus there.

  13. 4 out of 5

    Andrii Zakharov

    An absolute joy to read, this book just might make you fall in love with calculus. Yes, Steven Strogatz really is that good.

  14. 4 out of 5

    Steve

    Great overall explanation of calculus by looking at its history. You won’t learn calculus from this book, but you do learn ABOUT it. I had first-semester calculus many decades ago, I’ve forgotten it all and this book didn’t bring it back to me either. But it was fun to read. My eyes certainly glazed over for some pages here or there, but the vast majority is very readable even if you had zero math background I think.

  15. 4 out of 5

    Jason Furman

    A fantastic book about calculus. A blend of the history of the development of calculus, its applications, and intuitive explanations of its power filled with nicely intuitive explanations that will either provide a refresher or a different way of understanding what you have already learned. Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back A fantastic book about calculus. A blend of the history of the development of calculus, its applications, and intuitive explanations of its power filled with nicely intuitive explanations that will either provide a refresher or a different way of understanding what you have already learned. Steven Strogatz proceeds in (sort of) chronological order, defining calculus not as what you learn in school but any technique that breaks things apart into infinitesimal pieces and puts them back together again in order to solve problems. Rather than describing an immaculate conception of calculus by Leibniz and Newton, Strogatz starts with Archimedes, shows several geometric applications, and even spends a lot of time on Descartes and Fermat before even getting to what we consider calculus today. In all of these he shows how a combination of abstract ideas but also in many cases practical problems led to the development of calculus. The chronological order is interrupted (in a good way) by Strogatz’s many descriptions of the applications of calculus to different practical problems, most of which are in the analytically relevant chapter. These include GPS, AIDS drugs, rocketry, and more. In all of these cases Strogatz shows his pedigree as an applied mathematician, going into significant but highly readable detail about the models and discoveries underlying these areas. Overall, the book is very nicely written and highly recommended.

  16. 4 out of 5

    Gary Moreau

    This is not intended to be a textbook on calculus. And, like a lot of calculus itself, it is and it isn’t - quite. It is, however, a book about the history of calculus, which is fascinating, and the degree to which the universe seems to have been coded in a way that calculus seems to have an uncanny ability to explain is, well, somewhat inexplicable. But as the author notes in the beginning, “For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that w This is not intended to be a textbook on calculus. And, like a lot of calculus itself, it is and it isn’t - quite. It is, however, a book about the history of calculus, which is fascinating, and the degree to which the universe seems to have been coded in a way that calculus seems to have an uncanny ability to explain is, well, somewhat inexplicable. But as the author notes in the beginning, “For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it’s the only way a universe with us in it could be, because nonmathematical universes can’t harbor life intelligent enough to ask the question.” How often do you hear a Professor of Applied Mathematics, at an Ivy League school no less, say something even remotely so self-reflective? Steven Strogatz is a great communicator who is both a great mathematician and who, it is easy to tell, gets goose bumps every time he thinks about the wonders of calculus. I am not a professional mathematician but have always found mathematics to be both fascinating and, well, not easy, but very relatable. It’s predictable, and that’s comforting once you can see the pattern. If you don’t feel quite that way you may – spoiler alert – find this book to be a bit more like a textbook than advertised. There are plenty of equations and symbols and the like. That is, after all, the alphabet of calculus. But here’s the thing. Unless you are also a math professor, you can ignore all of that. Just go with the prose. It tells the same story, but in a far more relatable form to the average lover of the written word. Just ignore the symbols. If you do you will miss nothing and you will find the professor’s enthusiasm to be quite contagious. The beauty of the book is that it is written from a perspective of humility. Both in terms of the enormity of calculus (Most people will relate to the subject matter simply as science.), and in terms of how far we have yet to go in terms of truly understanding the universe and the reality that defines it. We’ve only explained the tip of the iceberg. Math is a human convention. It’s not hydrogen or oxygen. It’s not even dark matter, which we “know” makes up most of the universe but which no one has ever isolated, although the Chinese are close. It is very accurate at deciphering reality if getting close to the “real” explanation is close enough. But close is only close. It isn’t reality itself. Reality is, after all, by definition, real. That is, ultimately, the problem with the promise of AI. Because AI is ultimately dependent on calculus and other disciplines of mathematics, it will get very smart, but it will never be human. What it will do, however, if we let it, is dumb down what it means to be smart to a standard perfectly suited to its abilities but ignorant of its shortcomings. That’s why, despite the promises of the silicon gods, we are very unlikely to see fully autonomous vehicles for decades to come. The only way that could happen is if we take all human drivers off the road overnight (The AI isn’t the problem; it’s us. We are unpredictable.), switch every vehicle to an autonomous vehicle all at the same time, and rebuild our infrastructure to accommodate the vulnerabilities of the various disciplines of mathematics on which the technology is based. And that’s obviously not going to happen. Nor do we want it to. Pi, as but one example, despite what you were taught in school, is not a number; it’s a range. It’s a small range, to be sure, but it’s a range nonetheless. In other words, it is precise enough for most things, but it is NOT the fabric of the universe. Science is a methodology for understanding reality; it is not, in the most literal sense, reality itself. Reality is not “waiting” to be discovered. It is. And just as an artist can draw a landscape, science can draw reality. Neither, however, IS reality. The history of calculus is truly fascinating. And that, to me, as a reader, makes it entertaining. Newton and Galileo and all the rest were truly amazing people. It boggles the mind to think of what they concluded when they did. Perhaps the book’s greatest contribution, however, is that it will put Silicon Valley in perspective. You may think your smart phone has changed your life in ways that nothing else possibly could. You’re wrong. I am a great admirer of Steve Jobs but James Clerk Maxwell (a Scot in the 1860s) changed your life in ways that Steve did not come close to. And that is why this book is so timely. Calculus is changing our world, and not entirely in good ways. If ever we needed perspective we need it now. Math is elegant. It was designed that way. (Remember that it is not of the universe, like rain or sunshine.) And it does have an uncanny reliability that helps us to understand the world around us. Take GPS. We all use it. We all rely on it. But did you know that GPS is all about time, not navigation? Those GPS satellites don’t “see” you; they time you. It only works because scientists came to understand the mathematics of what we call time at a very precise level. That’s not reality, of course, because time is a concept (time, even as we understand it, varies with altitude), but it is close enough to give us GPS. And isn’t that an amazing thing. I think so. And that’s why I found this to be such an enjoyable book, beyond the fact that I am simply stimulated by really enthusiastic people and Professor Strogatz is one of the most enthusiastic people I have had the privilege to read in a really long time. If, on the other hand, you prefer a good murder mystery, or something with a little romance, at least, you won’t find it in this book. But that’s just my opinion. A little like pi, if you will. Pretty accurate, but not reality itself. Decide for yourself. You won’t be wasting your time.

  17. 5 out of 5

    David Schwartz

    Well written and entertaining look at the development of calculus - surely one of mankind's most impressive intellectual achievements - and how it comes into play in all sorts of important problem solving. You meet Archimedes, Descartes, Fermat, Newton, Leibniz, and sundry other geniuses along the way - even the women who figured in the Hidden Figures story. Once you've finished this, you owe it to yourself, if you have not done so, to take a full course or work through a good textbook on your o Well written and entertaining look at the development of calculus - surely one of mankind's most impressive intellectual achievements - and how it comes into play in all sorts of important problem solving. You meet Archimedes, Descartes, Fermat, Newton, Leibniz, and sundry other geniuses along the way - even the women who figured in the Hidden Figures story. Once you've finished this, you owe it to yourself, if you have not done so, to take a full course or work through a good textbook on your own. I would recommend some of the older, less expensive textbooks - I have one from my father-in-law written by Lyman Kells, and it's great. Also check out this website for on line texts: http://onlinebooks.library.upenn.edu/...

  18. 5 out of 5

    Melissa Dee

    Just to set the stage, I’m a math geek, and I do speak calculus, if somewhat imperfectly. I was thrilled by this book. Strogatz explores “the calculus” from its beginnings to the outer reaches of its applications and unsolved problems of today. He gives a very accessible explanation of both Newton’s and Leibniz’s approach to the development of the mathematics now called calculus. For the reader who doesn’t want to follow along with the math, “Infinite Powers” is still a very readable Just to set the stage, I’m a math geek, and I do speak calculus, if somewhat imperfectly. I was thrilled by this book. Strogatz explores “the calculus” from its beginnings to the outer reaches of its applications and unsolved problems of today. He gives a very accessible explanation of both Newton’s and Leibniz’s approach to the development of the mathematics now called calculus. For the reader who doesn’t want to follow along with the math, “Infinite Powers” is still a very readable glimpse of the strength and beauty of the mathematical concepts. "Infinite Powers” will be tremendously useful in explaining to the high school students I teach why it is that calculus should make their hearts sing! I voluntarily read and reviewed an advanced copy of this book. All thoughts and opinions are my own.

  19. 4 out of 5

    Chunyang Ding

    A very well written introduction to calculus, in a refreshingly different perspective than how it typically is presented. Quite enjoyed it, but there were some stories that seemed more repetitive than was needed. It is very well suited for the target audience of people who are interested in calculus and had not had a chance to learn it in the past. Steven Strogatz is a wonderful writer, although I thought that Sync was more engaging (as it was based on his own research).

  20. 4 out of 5

    Connor Oswald

    I get why mathematicians hate physicists better now.

  21. 4 out of 5

    Truly

    Watch a detailed review along with my favorite ideas and takeaways at: https://youtu.be/5WOvVSTdUmY

  22. 5 out of 5

    Daniel

    This book put me off part way through the first chapter as the author waxed poetic about calculus as the center of the universe and "God's language" (according to Feynman with whom I previously took issue). Since my dad was reading it though, I read further, and was rewarded - at least initially. At its best, this book tells some fascinating history surrounding major mathematical figures and their circumstances, which is a joy to read. I particularly enjoyed reading about contemporaries Des This book put me off part way through the first chapter as the author waxed poetic about calculus as the center of the universe and "God's language" (according to Feynman with whom I previously took issue). Since my dad was reading it though, I read further, and was rewarded - at least initially. At its best, this book tells some fascinating history surrounding major mathematical figures and their circumstances, which is a joy to read. I particularly enjoyed reading about contemporaries Descartes and Fermat, and their resulting rivalry. More typically, it provides a gentle and approachable introduction by way of analogies to the core concepts of calculus. At its worst, and sadly far too often, it bludgeons the reader with proclamations of calculus as the most important thing simply because it has been used to solve a lot of real world problems (so has writing). Combined with a writing style reminiscent of the first edict provided to a child on essay writing: "tell them what you're going to tell them, tell them, and finally tell them what you told them," this book became mightily cumbersome by the end. Despite some interesting tidbits and stories, it was work to finish, and not a particularly rewarding type either.

  23. 5 out of 5

    Chen Xin

    Shear elegance of writing! Clarity, simplicity, rigor, warmth...... Wow

  24. 4 out of 5

    Laura

    Totally weird to get chills reading a math book I know but can’t help it. Taking calculus in HS was life changing for me. Because of a good teacher (Mr. Olsson!) I saw the magic and beauty of mathematics on full display. Yet I struggle sometimes to give a succinct answer to “what is calculus?” Language of the universe, study of change, ...ok ...but I think there must be better answers. And so far I have confidence Mr. Strogatz has some... Archimedes Palimpsest now at the Walters Art M Totally weird to get chills reading a math book I know but can’t help it. Taking calculus in HS was life changing for me. Because of a good teacher (Mr. Olsson!) I saw the magic and beauty of mathematics on full display. Yet I struggle sometimes to give a succinct answer to “what is calculus?” Language of the universe, study of change, ...ok ...but I think there must be better answers. And so far I have confidence Mr. Strogatz has some... Archimedes Palimpsest now at the Walters Art Museum in Baltimore. The only surviving copy of Archimedes Method. It had disappeared until 1998, under a liturgical text. Seesaw, parabola, triangle, slices.

  25. 5 out of 5

    Eugenio Gomez-acebo

    Interesting and accessible history of calculus, from integral to differential equations. It is a bit extreme to describe calculus as the "language of God" and how calculus becomes modern, advances and changes our universe as a self-conscious entity. There is certain magic in calculus, but there is more in how humans have used this tool to uncover the secrets of our universe. But I guess you need to add some drama to sell a book. In any case, it is fascinating to discover the way Archimedes estim Interesting and accessible history of calculus, from integral to differential equations. It is a bit extreme to describe calculus as the "language of God" and how calculus becomes modern, advances and changes our universe as a self-conscious entity. There is certain magic in calculus, but there is more in how humans have used this tool to uncover the secrets of our universe. But I guess you need to add some drama to sell a book. In any case, it is fascinating to discover the way Archimedes estimated the area of a circle, the three Zenon paradoxes, how to estimate the value of pi, the area of a parable sector, the retrograd movement of Mars, the laws of a pendulum, how longitude is estimated, the three Kepler laws, how Fermat invented the cartesian coordinates (and not Descartes) and the optimization techniques that we use today to compress signals...

  26. 5 out of 5

    William Bies

    Steven Strogatz is an applied mathematician at Cornell University who has risen to prominence in the past decades as a communicator with the general public, what many would suppose to be a thankless task owing to the poor knowledge of mathematics on the part of the typical person these days, even when educated, and to the inherent difficulty of conveying what mathematicians prize in their work. Strogatz, though, is eager to take on the challenge. His background as a noted researcher on dynamical Steven Strogatz is an applied mathematician at Cornell University who has risen to prominence in the past decades as a communicator with the general public, what many would suppose to be a thankless task owing to the poor knowledge of mathematics on the part of the typical person these days, even when educated, and to the inherent difficulty of conveying what mathematicians prize in their work. Strogatz, though, is eager to take on the challenge. His background as a noted researcher on dynamical systems and chaos theory (he has written a leading textbook in the field) outfits him well for the task, as it has acquainted him with a kind of mathematics that may be especially accessible to the layman. His most recent contribution, Infinite Powers, devotes itself to the calculus invented during the seventeenth century, which formed the groundwork of the early modern scientific revolution. Now, Strogatz has the disconcerting habit of using the word in an expansive sense, to mean everything comprised under what one would ordinarily refer to with the general term, analysis, but what’s in a name? Clearly, the calculus properly speaking was the driving force behind the development of modern analysis and mathematical physics. Strogatz does have a knack for putting complicated things into simple terms (perhaps too simple, for this reviewer). The curious reader will enjoy following his account of the history of mathematical ideas as they apply to calculus, embellished with elaborate explanations for the rank beginner. Those who know something about the material can read at another level, quickly but attuned to the author’s apt way of putting familiar, or in some cases, perhaps not-so-familiar things. One can discern the hand of the born teacher. This reviewer was pleased, for instance, by Strogatz’s retelling of what Newton means by fluents and fluxions, his observations on why integral calculus is harder than differential calculus (the former reduces to a global, the latter to a local problem) and his reconstruction of the historical path by which the young Newton solved integration by playing with power series. His summary of Leibniz’s career and comparison with Newton’s is also quite good. This book would be worth reading for a sole point, what Strogatz somewhat infelicitously terms the ‘infinity principle’. The problem-solving power of the calculus flows from its method of analysis and synthesis, so adroitly wielded by Newton; namely, one breaks the problem down into infinitely many small pieces, where it is easiest to analyze, then puts all the pieces back together again to yield the solution. Strogatz does a nice job of illustrating his infinity principle, in terms readily perceptible to the amateur, in the work of Archimedes, among others. Now, to some disappointments: Strogatz is not by any means an intellectual historian and, whereas his skillful presentations of elementary mathematics are polished and presumably helpful to learners, his remarks on general issues in the history of ideas are amateurish and do not go beyond the stereotypical. He has, apparently, read many, if not all, of the primary sources in mathematics and physics, but can scarcely have engaged in serious study of the history of philosophical or religious movements. Hence, one will miss in him the perspective on and incisive observations into the context and motivation of the scientific discoveries, such as one could expect as a matter of course from an eminent scholar such as Ernst Cassirer. Consult Strogatz for strokes of clarity in the exposition of technical matters and anecdotal history, little else, nothing having to do, say, with the genesis of ideas or epistemology or methodology in the philosophy of science. It will be best if Strogatz limit himself to his domain of expertise, in applied mathematics and allied scientific disciplines. This last comment prompts another reflection. The technological and engineering advances characteristic of our day were made possible by the advent of the calculus and Strogatz is in his native element when recounting them. Based on his curriculum vitae, however, as an applied mathematician, Strogatz cannot, in all likelihood, possess very much technical competence in the loftier realms of modern mathematics, such as abstract algebra, algebraic topology and geometry, differential geometry, functional analysis and so forth. These latter subjects constitute the research frontier and underlie the remarkable confluence between higher mathematics and mathematical physics that has emerged, meteorically, over recent decades. From what one can tell, the educated public would be keenly interested in participating as spectators in the developments taking place in fundamental physics, and the attendant controversy over the role of string theory versus loop quantum gravity or noncommutative geometry etc. Now, in this reviewer’s estimation, it is here only that the sheer intelligible beauty of mathematics comes to light and that scope is afforded to the genial imagination and inventiveness of the artist. The moral lesson: Strogatz can get one only so far, but is bound by his commitment to basic material to remain innocent of the most profound and wonderful aspects to be contemplated in the world of higher mathematics. If only some of those at the forefront of the state of the art would apply themselves to communication with the larger public, as Strogatz has faithfully done! Perhaps Edward Frenkel’s award-winning Love and math: The heart of hidden reality answers to what this reviewer is looking and hoping for!

  27. 4 out of 5

    Mike

    I found the beginning of this book peomising as the author made calculus sounds incredibly intriguing. However, I found this book to not make calculus that much more interesting than I’d thought befure due to how shallowly the author penetrates the subject. Additionally, I found the ending of this book to be a drag due to the author forgoing any mathematical explainations and merely describing complicated processes that calculus was magically the answer to. This effectively sucked any of the int I found the beginning of this book peomising as the author made calculus sounds incredibly intriguing. However, I found this book to not make calculus that much more interesting than I’d thought befure due to how shallowly the author penetrates the subject. Additionally, I found the ending of this book to be a drag due to the author forgoing any mathematical explainations and merely describing complicated processes that calculus was magically the answer to. This effectively sucked any of the intrigue out of these otherwise fascinating situations. Sprinkle in a bunch of half-baked analogies and the end of this book is fairly boring and tedious.

  28. 5 out of 5

    Lucas Wiman

    I mostly bought this book because I've found Strogatz interesting on Twitter and in interviews, and I tend to buy people's books who fit that description, if only to support them and their work. I probably wouldn't have read this book if I hadn't already bought it, but I'm glad I did. It's very well written, and helps to give a feeling of both the power and joy of calculus. It is not the sort of book you'd read to actually learn much calculus, but more to get excited about the topic a I mostly bought this book because I've found Strogatz interesting on Twitter and in interviews, and I tend to buy people's books who fit that description, if only to support them and their work. I probably wouldn't have read this book if I hadn't already bought it, but I'm glad I did. It's very well written, and helps to give a feeling of both the power and joy of calculus. It is not the sort of book you'd read to actually learn much calculus, but more to get excited about the topic and get some intuition. This would be a good book for interested students to read on break before they start their first calculus course. I think it does a much better job than most calculus textbooks of explaining both why the fundamental theorem of calculus is important and why it's true. It also does a good job of explaining what kinds of problems calculus lets you tackle with relative ease. As an example, the chapter on the development of the "cocktail" three-drug treatment for HIV was very inspiring and clear. With a few measurements and some differential equations, Strogatz walks the reader through research that led to a deeper understanding of how HIV interacts with the immune system and how combinations of drugs made the evolution of resistance much less likely. A few equations and calculus helped to transformed a disease that was a death sentence into a manageable chronic health problem, saving literally millions of lives. When I've used math to solve (much less important) practical problems in my career as a software developer, I've felt the same sort of rush of excitement and understanding that Strogatz very effectively puts forward to the reader here. I have an MS in math and I've known at least the basics of calculus since I was a freshman in high school, so little of the mathematical material was new to me. I personally would've found the book more interesting if it were more technical. For example, I'd have loved to learn how the modeling for CT scanning worked, which Strogatz only briefly touches on. Still, there was new and interesting material even for me, e.g. on the historical feud between Descartes and Fermat and many of the applications were new to me. I'll probably recommend this book to anyone who asks me for a recommendation to better understand calculus. One final quibble: I think the treatment of infinitesimals is a bit outdated. Strogatz treats infinitesimals (and differentials) as useful fictions, that are good for reasoning to a solution, but plagued with paradoxes and don't "really" exist. Yet in the 1960's logician Abraham Robinson created something called "nonstandard analysis", which shows that infinitesimals are consistent and can generate exactly the same results as "standard" analysis. Strogatz has this to say: Within the ideal world of mathematics, infinitesimals don’t exist in the real number system, but they do exist in certain nonstandard number systems that generalize the real numbers. For Leibniz and his followers, they existed as fictions of the mind that came in handy. That’s the way we will be thinking about them. I can see not wanting to get bogged down in nonstandard models of Peano arithmetic (which is how nonstandard analysis books tend to start), but Strogatz continues to treat infinitesimals as useful nonsense, despite apparently being aware they can be formalized just as rigorously as the Weierstrauss ε/𝜹 formalism. And the construction itself isn't that esoteric or weird. It's basically this: suppose there is a number c that is bigger than 1, 2, 3, 4, ..., but still smaller than infinity. It sounds paradoxical, but it's actually not. Then 1/c is smaller than 1/2 or 1/3 or 1/4 or ... but still bigger than zero. 1/c is an infinitesimal. That's basically it, the hard part is showing that everything works. With that small amount of exposition, a lot of paradoxes melt away. There's no need to get into the details, but treating infinitesimals as inherently paradoxical then using them throughout the book is just a strange way to approach things.

  29. 4 out of 5

    Grrlscientist

    Calculus is one the most profound inventions in human history. It underlies most modern technologies such as radio, television, radar, GPS navigation, cell phones, and MRI imaging. It informs meteorology, economics, social sciences, epidemiology, biology and medicine. Thus, it is a subject that every educated person should have a conceptual understanding of, at least the very least. In Steven Strogatz’s beautifully-written Infinite Powers: The Story of Calculus — The Language of the Calculus is one the most profound inventions in human history. It underlies most modern technologies such as radio, television, radar, GPS navigation, cell phones, and MRI imaging. It informs meteorology, economics, social sciences, epidemiology, biology and medicine. Thus, it is a subject that every educated person should have a conceptual understanding of, at least the very least. In Steven Strogatz’s beautifully-written Infinite Powers: The Story of Calculus — The Language of the Universe (Atlantic Books, 2019), we learn about the conceptual framework upon which calculus is built, we learn about its humble beginnings and how it was developed throughout the ages to address specific challenges from determining the area of a circle to making sure your space craft sticks its lunar landing. Because calculus is a supremely human endeavor, we also learn about this discipline by learning about its people, starting with Zeno’s and Archimedes’ seminal insights, and following its development over more than three millennia by learning about critical contributions made by dedicated men and women who made it possible to study modern day abstract concepts such as chaos theory and artificial intelligence. In this fun book, we discover that the special genius of calculus is that it’s based upon breaking down complex, seemingly intractable problems into infinitely small, solvable, pieces, which then can be computed and tamed before reassembling them into the larger, hopefully much less scary, whole. But more than breaking things into infinitely tiny pieces, this strategy has made calculus into a powerful tool that has led to all sorts of technological advances and solutions to real-world problems — solutions that most people never hear about in their maths classes, but really should. For example, I was particularly fascinated by the lucid and fascinating discussion about how calculus played a role in the universal adoption of the triple drug cocktail to combat the maddening, seemingly insurmountable mutation rate of HIV. Calculus also provided critical insights into the nature of the puzzling asymptotic stage of the HIV infection where, it turns out, there is a long-running, albeit delicate, balance in the furious battle between production of new virus particles and their destruction by the immune system (pp. 218–225). Additionally, I was amused to learn how one can use the standing electromagnetic waves emitted by their microwave oven to calculate the speed of light with reasonable accuracy (pp. 262–264). On a more theoretical level, Professor Strogatz’s thoughts about why synthesis is so much more difficult than analysis for problem solving was illuminating (pp. 102–103), especially when thinking about the ease of learning how to use differential equations compared to integral equations, which are more intellectually challenging. In this book, infinity is its alpha and omega, its beginning and end, and that intellectual framing is both appropriate and aesthetically pleasing. For readers who don’t want to immerse themselves in the actual maths but who still want to appreciate the history and the reasoning that gave rise to calculus, this is the most accessible book on those topics published in many years. In recognition of that rare achievement, this brilliant book is one of six that are shortlisted for the prestigious Royal Society Insight Investment Science Book Prize for 2019. For serious students of calculus, Infinite Powers will give you the historical and conceptual grounding that is so often lacking in maths education these days, and in doing so, it could help you understand at a more intuitive level what the heck you are doing. NOTE: Originally published at Forbes on 30 September 2019.

  30. 4 out of 5

    Hans Trivedi

    This book provides a neat perspective on the historical development of calculus and how it has been useful, largely by breaking up challenging problems to ‘infinitely’ many smaller pieces and then stitching them back together again! The almost story like exploration of mathematicians throughout history was enjoyable and refreshing way to read this book. The key takeaway being that their perseverance with these problems and obsessive focus took calculus from its humble beginnings to it This book provides a neat perspective on the historical development of calculus and how it has been useful, largely by breaking up challenging problems to ‘infinitely’ many smaller pieces and then stitching them back together again! The almost story like exploration of mathematicians throughout history was enjoyable and refreshing way to read this book. The key takeaway being that their perseverance with these problems and obsessive focus took calculus from its humble beginnings to its formidable stature today. Many of the explanations are not too rigorous and arguably rely on the reader possessing some pre-existing knowledge on the subject matter. I can imagine it’s a very different read depending on how familiar you are with the key concepts. Some knowledge makes a better read. One thing that I found that set this book apart was that it successfully focuses on giving a deeper intuitive perspective on understanding how mathematics is successfully applied. Providing a clear connection between calculus’ development and how it is applied to solving real world problems, leads to a clear understanding of how calculus truly works. This shines light on the fundamental “credo” of calculus; “To solve a hard problem about anything continuous, slice it into infinitely many parts and solve them. Then by putting the answers back together, you can make sense of the original whole.” Which the author aptly names ‘The Infinity Principle’. Roll credits. A key idea I found particularly illuminating stemmed from the carefully discussion of the difference between analysis and synthesis, with a focus on why (or how) one is much harder than the other. There seems to be a deep connection with this throughout the book, and between differentiation and integration themselves. More analysis and more depth across more problems would have been nice. Highly recommended read with a slightly bittersweet ending as you are left wishing the author wrote a longer book!

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